Clusters, filaments, sheets, and voids are the building blocks of the cosmic web. Forming dark matter halos respond to these different large-scale environments, and this in turn affects the properties of galaxies hosted by the halos. It is therefore important to understand the systematic correlations of halo properties with the morphology of the cosmic web, as this informs both about galaxy formation physics and possible systematics of weak lensing studies. In this study, we present and compare two distinct algorithms for finding cosmic filaments and sheets, a task which is far less well established than the identification of dark matter halos or voids. One method is based on the smoothed dark matter density field and the other uses the halo distributions directly. We apply both techniques to one high-resolution N-body simulation and reconstruct the filamentary/sheet like network of the dark matter density field. We focus on investigating the properties of the dark matter halos inside these structures, in particular, on the directions of their spins and the orientation of their shapes with respect to the directions of the filaments and sheets. We find that both the spin and the major axes of filament halos with masses 10¹³ h ^–1 M _☉ are preferentially aligned with the direction of the filaments. The spins and major axes of halos in sheets tend to lie parallel to the sheets. There is an opposite mass dependence of the alignment strength for the spin (negative) and major (positive) axes, i.e. with increasing halo mass the major axis tends to be more strongly aligned with the direction of the filament, whereas the alignment between halo spin and filament becomes weaker with increasing halo mass. The alignment strength as a function of the distance to the most massive node halo indicates that there is a transit large-scale environment impact: from the two-dimensional collapse phase of the filament to the three-dimensional collapse phase of the cluster/node halo at small separation. Overall, the two algorithms for filament/sheet identification investigated here agree well with each other. The method based on halos alone can be easily adapted for use with observational data sets.